Completely and Incompletely Specified Logic Functions

What is a Logic Function?

In digital electronics and Boolean algebra, an expression that have some operations performed on the binary inputs to produce an output in the binary form is called a logic function. Logic functions are also called Boolean functions.

Depending on the specified output, there are two types of logic functions namely,

• Completely specified logic functions

• Incompletely specified logic functions

Now, let us discuss the completely and incompletely specified logic functions individually in detail.

Completely Specified Logic Function

A logic function or Boolean function which is defined for all possible combinations of input variables is known as a completely specified logic function. In other words, if the output of a Boolean or logic function is known for all possible combinations of its inputs, then it is called a completely specified logic function.

For example, the logic expression of the output of a two input AND gate is an example of a completely specified logic function. This is because, for a two input AND gate, the output of the AND gate is defined for every possible combination of input variables.

Consider the truth table of a two input AND gate as given below −

Inputs		Output
A	B	Y
0	0	0
0	1	0
1	0	0
1	1	1

The logic expression for the output of the two input AND gate can be written directly from this truth table as follows,

$$Y\, =\,A\cdot B $$

From the above truth table, it can be seen that the output of the AND gate is 1 for the combination of inputs where both inputs are 1, and it is 0 for all other combinations of inputs.

In digital circuit design, the completely specified logic functions play a vital role because they ensure the well-defined and predictable operation of a digital circuit. If a digital circuit is designed with a logic function which is not completely specified, it might cause errors in the result.

Now, let us understand the incompletely specified logic function in digital electronics.

Incompletely Specified Logic Function

A logic function or a Boolean function whose output is not defined for every possible combinations of input variables is called an incompletely specified logic function.

In other words, when the output of a logic function or a Boolean function is not known or determined for at least one combination of inputs, then it called an incompletely specified logic function.

The expression of output of a two input XOR gate is an examples of an incompletely specified logic function. To know the output of a two input XOR gate for different possible combinations, consider its truth table given below −

Inputs		Output
A	B	Y
0	0	0
0	1	1
1	0	1
1	1	0

Thus, the Boolean expression or logic function of the output is given by,

$$Y\, =\,\bar{A} B+A \bar{B} $$

Hence, from the above truth table, it is clear that the output of the XOR gate is 1 if one of the inputs is 1, and the output is 0 if both inputs are the same. Although, the output of the XOR gate is not specified for the input combination 00, i.e. for input combination 00, it could be 0 or 1.

Incompletely specified logic functions are used in implementation of such digital circuits that represent logic functions which are not completely defined. They are also used to minimize complex logic functions by ignoring certain combinations of inputs. However, the incompletely specified logic functions can cause errors and unpredictable behavior in digital circuits.

Conclusion

In conclusion, a completely specified logic function is one whose output is known for every possible combinations of inputs, while the incompletely specified logic function is one whose output is not known at least for one input combination.